# Position vectors

1. Jun 15, 2010

### geoff18

1. The problem statement, all variables and given/known data
Let A,B,C be the three points in R^3 with position vectors
a=(1,-2,0)
b=(-1,1,2)
c=(7,1,6)

respectively.

Find the position vector q of the point Q that divides the line segment AB in the ratio 2:1, (where Q is closer to B.)

2. Relevant equations

3. The attempt at a solution
i have no idea how to start...

2. Jun 15, 2010

### geoff18

do i find the length of AB first and work out the ratio?

3. Jun 15, 2010

### HallsofIvy

Staff Emeritus
No, it is not necessary to find the length of AB! You can, for instead, draw horizontal and vertical lines making a right triangle having vertices at A and B. Then "similar triangles" will make your job easy. Just divide the horizontal and vertal lines in to 3 parts.

4. Jun 15, 2010

### geoff18

i tried to do that, but i have no idea how to find the position vector of q...

5. Jun 15, 2010

### geoff18

im waiting online for help, so any help is much appreciated.. thanks in advance. =)

6. Jun 15, 2010

### Staff: Mentor

Write the vector equation for the line L that contains the segment AB. This equation will be r(t) = a + t*v

In this equation, a is the vector from the origin to point A, and v is the vector from point A to point B. t is the parameter, and r is a vector that goes from the origin to the point on the line L determined by the parameter t.

What vector is represented by r(0)? By r(1)? Can you think of a way to get to a point 2/3 the way along the segment AB?

7. Jun 15, 2010

### geoff18

when r intersects the line?

8. Jun 15, 2010

### Staff: Mentor

Each point of the line corresponds to r(t) for some value of t.

9. Jun 15, 2010

### geoff18

so do i have to find t?
how do i find t?

10. Jun 15, 2010

### Staff: Mentor

You get to pick t. If I choose t = 0, r(0) = a + 0*v = <1, -2, 0>. This vector goes from the origin to point A. What I'm calling v is the vector from A to B. Since I am multiplying by 0, I don't need to do any calculations with v for this value of t.

What is r(1)?

11. Jun 17, 2010

### Susanne217

Best way to show this is to show there exists a socalled linear combination from Linear Algebra of all three vectors which satisfies that condition.

if the vector are called $$v_1,v_2,v_3$$

then a linear combination is $$u_1 \cdot v_1 + u_2 \cdot v_2 + u_3 \cdot v_3$$

where the $$u_1,u_2,u_3$$ are weights...

12. Jun 17, 2010

### HallsofIvy

Staff Emeritus
Since you apparently did not understand my first response, using "similar triangles" based on the coordinate axes:
The change in x-coordinate form a(1, -2, 0) to b(-1, 1, 2) is -1- 1= -2. 2/3 of that is -4/3. That is, the change in x coordinate from a(1, -2, 0) to point Q is -4/3: the x coordinate of Q is 1+ -4/3= -1/3.

Do the same for the y and z coordinates.